Here we focus on recurrence and transience of symmetric Markov processes
generated by Dirichlet forms.
Roughly speaking, we mean by recurrence the nature of particles
hitting every open set infinitely many times almost surely.
On the other hand,
we mean by transience the nature of particles escaping to infinity eventually almost surely.
There are many recurrence criteria
for symmetric Markov processes
in terms of the analytic and geometric conditions as mentioned before
(see, for instance, [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]});
however, there are less such transience criterion
in spite of their importance.
